When I asked the first question, people tended to tell me their model of the thing they were focusing on. I found this surprising, since it seemed like the most important part of their model ought to be the part that indicates why their project is more promising to focus on than the alternatives. Because of this, I began asking people what they were ignoring.

It makes sense that people would have a lot to say about things inside their area of focus. You’ll spend the most time thinking about the thing you are working on, not the thing you’ve decided isn’t worth your time.

But the basic act of narrowing down to one an area of focus is probably the single most important decision one can make. Promoting one area to your attention for further study already implies a substantial amount of judgment about the world. And since I want to compare different focus areas, and the arguments for them, before choosing one, it is this initial judgment that’s most relevant to me. And it is here that it looks as if little public work has been done.

Positive and disjunctive deductive systems

Near the beginning of Euclid’s Elements, Euclid lays out five premises commonly called “Postulates”, the fifth of which is:

That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Mathematicians long suspected that something was a bit funny about this postulate. At first, they thought this might be because the fifth postulate could be proved on the basis of the other axioms in the book. But eventually mathematicians such as Lobachevsky and Bolyai figured out that it was the opposite. Not only could the fifth postulate not be proved, but there could be consistent geometries in which it was false!

This was hard to see because Euclid lays out geometry in a primarily positive way - the axioms are taken to be true, and then we see what follows from them. Positive deductive systems are easy to keep track of, and can be used to generate a set of propositions which are all true at the same time, and necessarily cohere.

Another way to lay out the fifth postulate might have to take the fifth postulate and situate it in a broader of possibilities. This is easier to do using the more common equivalent formulation known as the parallel postulate:

Given any straight line and a point not on it, there "exists one and only one straight line which passes" through that point and never intersects the first line, no matter how far they are extended.

If Euclid had wanted to make it clearer what constraints he was accepting in his geometric system, he might instead have written:

Given any straight line and a point not on it, a line might conceivably be drawn through that point such that, if extended indefinitely in either direction, it will never meet the first line. The number of such lines that can actually be drawn is either:

None

One

Infinitely many

Let it be the case that exactly one such line can be drawn and this line be called parallel to the first.

Reading the above disjunction, it would be easy for a geometer to intuit that there are two additional domains to be explored, which might or might not be productive.

In general, when a deductive argument makes disjunctions like the one above explicit, they are disjunctive, while those deductive arguments that only make clear what is true inside the system I call positive.

An excellent example of a disjunctive argument around existential risk is Nick Bostrom’s simulation argument, which I’ve referred to before:

This paper argues that at least one of the following propositions is true:

the human species is very likely to go extinct before reaching a “posthuman” stage;

any posthuman civilization is extremely unlikely to run a significant number of simulations of their evolutionary history (or variations thereof);

we are almost certainly living in a computer simulation.

It follows that the belief that there is a significant chance that we will one day become posthumans who run ancestor-simulations is false, unless we are currently living in a simulation. A number of other consequences of this result are also discussed.

While disjunctive arguments are better at contextualizing claims, they may take more time and thought to specify correctly. Not every disjunction may be worth making, and we well talk about parts of arguments as being more or less disjunctive, rather than grouping everything into two categories.

Even so, I suspect that current discourse in many domains has vastly underinvested in making disjunctions explicit. If we want to persuade people who do not already agree with us, not with charm or rhetorical force but with arguments - if we want to leave ourselves open to being persuaded that we ourselves are wrong, then we need to make the basic premises of our arguments explicit.

[…] arguments are not all mutually exclusive, but each one seems to stand on its own. In the spirit of making disjunctions explicit, I will treat each one separately, so I can assess the strengths, weaknesses, and implications of […]